# The Weyl-Wigner-Moyal formalism on a discrete phase space. I. A Wigner function for a nonrelativistic particle with spin

## TL;DR

This paper extends the Weyl-Wigner-Moyal formalism to quantum particles with discrete internal degrees of freedom, providing explicit formulas for Wigner functions and quantizers for systems with arbitrary spin.

## Contribution

It develops a formalism linking operators to phase space functions for particles with spin, including explicit expressions for Wigner functions and quantizers for arbitrary spin values.

## Key findings

- Derived a one-to-one correspondence between Hilbert space operators and phase space functions.
- Explicit formulas for the Stratonovich-Weyl quantizer, star product, and Wigner functions for systems with spin.
- Analyzed Wigner functions for Landau levels and magnetic resonance of spin-1/2 particles.

## Abstract

The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space $L^{2}(\mathbb{R}^{3})\otimes{\mathcal{H}}^{(s+1)}$ and functions on the phase space $\mathbb{R}^{3}\times\mathbb{R}^{3}\times \{0,...,s\} \times\{0,...,s\}$ is found. The expressions for the Stratonovich-Weyl quantizer, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin $\frac{1}{2}$ nonrelativistic particle as well as the magnetic resonance for a spin $\frac{1}{2}$ nonrelativistic uncharged particle are analysed.